Approximation of Generalized Ovals and Lemniscates towards Geometric Modeling

This paper considers an approach towards the building of new classes of symmetric closed curves with two or more focal points, which can be obtained by generalizing classical definitions of the ellipse, Cassini, and Cayley ovals. A universal numerical method for creating such curves in mathematical packages is introduced. Specific aspects of the provided numerical data in computer-aided design systems with B-splines for three-dimensional modeling are considered. The applicability of the method is demonstrated, as well as the possibility to provide high smoothness of the curvature profile at the specified accuracy of modeling.

A Study on the Errors of 2D Circular Trajectories Generated on a 3D Printer

This paper presents a study on the movement precision and accuracy of an extruder system related to the print bed on a 3D printer evaluated using the features of 2D circular trajectories generated by simultaneous displacement on x and y-axes. A computer-assisted experimental setup allows the sampling of displacement evolutions, measured with two non-contact optical sensors. Some processing procedures of the displacement signals are proposed in order to evaluate and to describe the circular trajectories errors (e.g., open and closed curves fitting, the detection of recurrent periodical patterns in x and y-motions, low pass numerical filtering, etc.). The description of these errors is suitable to certify that the 3D printer works correctly (keeping the characteristics declared by the manufacturer) for maintenance purpose sand, especially, for computer-aided correction of accuracy (e.g., by error compensation).

Drawing System with Dobot Magician Manipulator Based on Image Processing

In this paper, the robot arm Dobot Magician and the Raspberry Pi development platform were used to integrate image processing and robot-arm drawing. For this system, the Python language built into Raspberry Pi was used as the working platform, and the real-time image stream collected by the camera was used to determine the contour pixel coordinates of image objects. We then performed gray-scale processing, image binarization, and edge detection. This paper proposes an edge-point sequential arrangement method, which arranges the edge pixel coordinates of each object in an orderly manner and places them in a set. Orderly arrangement means that the pixels in the set are arranged counterclockwise to the closed curve of the object shape. This arrangement simplifies the complexity of subsequent image processing and calculation of the drawing path. The number of closed curves represents the number of strokes in the drawing of the manipulator. In order to reduce the complexity of the drawing of the manipulator, a fewer number of closed curves will be necessary. To achieve this goal, we not only propose the 8-NN (abbreviation for eight-nearest-neighbor) search, but also use to the 16-NN search and the 24-NN search methods. Drawing path points are then converted into drawing coordinates for the Dobot Magician through the Raspberry Pi platform. The structural design of the Dobot reduces the complexity of the experiment, and its attitude and positioning control can be accurately carried out through the built-in API function or the underlying communication protocol, which is more suitable for drawing applications than other fixed-point manipulators. Experimental results show that the 24-NN search method can effectively reduce the number of closed curves and the number of strokes drawn by the manipulator.

Membranes with thin and heavy inclusions: Asymptotics of spectra

We study the asymptotic behaviour of eigenvalues of 2D vibrating systems with mass density perturbed in a vicinity of closed curves. The threshold case in which the resonance frequencies of the membrane and the frequencies of thin inclusion coincide is investigated. The perturbed eigenvalue problem can be realized as a family of self-adjoint operators acting on varying Hilbert spaces. However the so-called limit operator is non-self-adjoint and possesses the Jordan chains of length 2. Apart from the lack of self-adjointness, the operator has non-compact resolvent. As a consequence, its spectrum has a complicated structure, for instance, the spectrum contains a countable set of eigenvalues with infinite multiplicity. The complete asymptotic analysis of eigenvalues has been carried out.

Shape-changing chains for morphometric analysis of 2D and 3D, open or closed outlines

Morphometrics is a multivariate technique for shape analysis widely employed in biological, medical, and paleoanthropological applications. Commonly used morphometric methods require analyzing a huge amount of variables for problems involving a large number of specimens or complex shapes. Moreover, the analysis results are sometimes difficult to interpret and assess. This paper presents a methodology to synthesize a shape-changing chain for 2D or 3D curve fitting and to employ the chain parameters in stepwise discriminant analysis (DA). The shape-changing chain is comprised of three types of segments, including rigid segments that have fixed length and shape, scalable segments with a fixed shape, and extendible segments with constant curvature and torsion. Three examples are presented, including 2D mandible profiles of fossil hominin, 2D leaf outlines, and 3D suture curves on infant skulls. The results demonstrate that the shape-changing chain has several advantages over common morphometric methods. Specifically, it can be applied to a wide range of 2D or 3D profiles, including open or closed curves, and smooth or serrated curves. Additionally, the segmentation of profiles is a flexible and automatic protocol that can consider both biological and geometric features, the number of variables obtained from the fitting results for statistical analysis is modest, and the chain parameters that characterize the profiles can have physical meaning.

Humans use minimum cost movements in a whole-body task

Humans have elegant bodies that allow gymnastics, piano playing, and tool use, but understanding how they do this in detail is difficult because their musculoskeletal systems are extraordinarily complicated. Nonetheless, common movements like walking and reaching can be stereotypical, and a very large number of studies have shown their energetic cost to be a major factor. In contrast, one might think that general movements are very individuated and intractable, but our previous study has shown that in an arbitrary set of whole-body movements used to trace large-scale closed curves, near-identical posture sequences were chosen across different subjects, both in the average trajectories of the body’s limbs and in the variance within trajectories. The commonalities in that result motivate explanations for its generality. One explanation could be that humans also choose trajectories that are economical in cost. To test this hypothesis, we situate the tracing data within a forty eight degree of freedom human dynamic model that allows the computation of movement cost. Using the model to compare movement cost data from nominal tracings against various perturbed tracings shows that the latter are more energetically expensive, inferring that the original traces were chosen on the basis of minimum cost.

Powers of Elliptic Scator Numbers

Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.

Vassiliev measures of complexity of open and closed curves in 3-space

In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities.

New Cubic Trigonometric Bezier-Like Functions with Shape Parameter: Curvature and Its Spiral Segment

This work presents the new cubic trigonometric Bézier-type functions with shape parameter. Basis functions and the curve satisfy all properties of classical Bézier curve-like partition of unity, symmetric property, linear independent, geometric invariance, and convex hull property and have been proved. The C 3 and G 3 continuity conditions between two curve segments have also been achieved. To check the applicability of proposed functions, different types of open and closed curves have been constructed. The effect of shape parameter and control points has been observed. It is observed that, by decreasing the value of shape parameter, the curve moves toward the control polygon and vice versa. The CT-Bézier curve is closer to the cubic Bézier curve for a fixed value of shape parameter. The proposed CT-Bézier curve can be used to represent ellipse. Using proposed basis functions, we have constructed the spiral segment which is very useful to construct fair curves and desirable to design trajectories of mobile robots, highway, and railway routes’ designing.